On explicit representations of isotropic measures in John and Löwner positions
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909463975297024 |
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| author | Baêta, F. M. Haddad, J. |
| author_facet | Baêta, F. M. Haddad, J. |
| contents | Given a convex body $K \subseteq \mathbb R^n$ in Löwner position we study the problem of constructing a non-negative centered isotropic measure supported in the contact points, whose existence is guaranteed by John's Theorem. The method we propose requires the minimization of a convex function defined in an $\frac {n(n+3)}2$ dimensional vector space. We find a geometric interpretation of the minimizer as $\left. \frac{\partial}{\partial r}(A_r, v_r)\right|_{r=1}$, where $A_r K + v_r$ is a one-parameter family of positions of $K$ that are in some sense related to the maximal intersection position of radius $r$ defined recently by Artstein-Avidan and Katzin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_03624 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On explicit representations of isotropic measures in John and Löwner positions Baêta, F. M. Haddad, J. Metric Geometry Differential Geometry 52A40, 52A05 Given a convex body $K \subseteq \mathbb R^n$ in Löwner position we study the problem of constructing a non-negative centered isotropic measure supported in the contact points, whose existence is guaranteed by John's Theorem. The method we propose requires the minimization of a convex function defined in an $\frac {n(n+3)}2$ dimensional vector space. We find a geometric interpretation of the minimizer as $\left. \frac{\partial}{\partial r}(A_r, v_r)\right|_{r=1}$, where $A_r K + v_r$ is a one-parameter family of positions of $K$ that are in some sense related to the maximal intersection position of radius $r$ defined recently by Artstein-Avidan and Katzin. |
| title | On explicit representations of isotropic measures in John and Löwner positions |
| topic | Metric Geometry Differential Geometry 52A40, 52A05 |
| url | https://arxiv.org/abs/2111.03624 |